Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposi...Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.展开更多
Circuit theory is an extremely important basic theory in electrical and electronic sciences and technologies. Over more than a century, researchers have come to the conclusion that a fundamental law of circuits needs ...Circuit theory is an extremely important basic theory in electrical and electronic sciences and technologies. Over more than a century, researchers have come to the conclusion that a fundamental law of circuits needs to satisfy the following three conditions: (1) Independency. It must be able to solve independently the basic problems of general solutions to the distribution of current and voltage in circuits. (2) Fundamentality. It cannot be derived from circuit theory and it must be the starting point for the establishment of circuit theory; it deduces the problem relevant to circuit theory by using purely logical inference, and establishes circuit theory into an independent deductive system. (3) Applicability. It must be widely applicable to all spheres of circuits, which includes sinusoidal steady-state linear and nonlinear networks, non-sinusoidal steady-state linear and nonlinear networks, transient-state processes, etc. From all networks to which the fundamental law of circuits applies, sinusoidal steady-state linear network is chosen as the most basic one to demonstrate that the two independent equations of circuits in integral form derived from Maxwell equations are able to meet these three conditions. Consequently, it is believed to be the fundamental law of circuits newly recognized today. This paper also makes the initiative to establish a circuit theory by which the basic rules of electromagnetic field govern the circuits, and the unity of electromagnetic fields and circuits is achieved.展开更多
文摘Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.
文摘Circuit theory is an extremely important basic theory in electrical and electronic sciences and technologies. Over more than a century, researchers have come to the conclusion that a fundamental law of circuits needs to satisfy the following three conditions: (1) Independency. It must be able to solve independently the basic problems of general solutions to the distribution of current and voltage in circuits. (2) Fundamentality. It cannot be derived from circuit theory and it must be the starting point for the establishment of circuit theory; it deduces the problem relevant to circuit theory by using purely logical inference, and establishes circuit theory into an independent deductive system. (3) Applicability. It must be widely applicable to all spheres of circuits, which includes sinusoidal steady-state linear and nonlinear networks, non-sinusoidal steady-state linear and nonlinear networks, transient-state processes, etc. From all networks to which the fundamental law of circuits applies, sinusoidal steady-state linear network is chosen as the most basic one to demonstrate that the two independent equations of circuits in integral form derived from Maxwell equations are able to meet these three conditions. Consequently, it is believed to be the fundamental law of circuits newly recognized today. This paper also makes the initiative to establish a circuit theory by which the basic rules of electromagnetic field govern the circuits, and the unity of electromagnetic fields and circuits is achieved.
基金厦门大学“基础创新科研基金(中央高校基本科研业务费专项资金)”资助Supported by the Fundamental Research Funds for the Central Universities)成果,项目编号:2011221032国家社科基金项目“国家利益视角下的国际法与中国应对策略研究”(10BFX090)阶段性成果
